the cap product
The cup product multiplies two cohomology classes. The cap product is the mixed operation that lets a cohomology class act on a homology class, lowering its degree: a cochain eats part of a chain and leaves the rest behind. It is the bridge that makes Poincare duality concrete — capping with the fundamental class is the very map that turns cohomology into homology on a manifold.
Concretely the cap product pairs H^p(X; R) with H_n(X; R) and produces a class in H_{n-p}(X; R). On chains: given a cochain alpha of degree p and a singular n-simplex sigma, alpha cap sigma is alpha applied to the front p-face of sigma, times the back (n-p)-face of sigma as a chain, alpha cap sigma = alpha(sigma restricted to [v_0, ..., v_p]) times (sigma restricted to [v_p, ..., v_n]). So the cochain consumes the front faces and what survives is a lower-dimensional chain. A Leibniz-type identity relating boundary, coboundary, cup and cap guarantees the operation descends to homology and cohomology classes, and the cap is compatible with the cup: (alpha cup beta) cap z = alpha cap (beta cap z).
The cap product's headline role is Poincare duality. On a closed oriented n-manifold M there is a fundamental class [M] in H_n(M), and capping with it, alpha -> alpha cap [M], is an isomorphism H^p(M) -> H_{n-p}(M). Thus the cap product is not a curiosity but the operational heart of duality: it is the explicit isomorphism, not merely an abstract claim that the groups are isomorphic. It also gives the projection formula and underlies the slant product and intersection theory.
A caution about variance and signs. The cap product is natural in a mixed way — it satisfies a projection formula f_*(f^* alpha cap z) = alpha cap f_*(z) rather than simple naturality, because cohomology is contravariant while homology is covariant. As with the cup product there are degree-dependent signs to respect. And capping reduces homological degree by the cohomological degree, never increases it; mixing up which factor is consumed (front versus back face) is a standard source of sign and degree errors.
On a closed oriented surface of genus g with fundamental class [M] in H_2, capping a degree-1 cohomology class alpha with [M] yields a degree-1 homology class alpha cap [M] in H_1, and this assignment is the Poincare duality isomorphism H^1(M) -> H_1(M). Geometrically, alpha is dual to a curve, and capping reads off that curve as a homology cycle.
Capping with the fundamental class realizes the Poincare duality isomorphism.
The cap product lowers homological degree by the cohomological degree and obeys a projection formula rather than plain naturality, because homology is covariant while cohomology is contravariant. Track which face (front versus back) each factor consumes, or signs and degrees go wrong.